The NCAA Division I Men's Basketball Championship is a single-elimination tournament that holds a unique place in the American sporting landscape. Sports have long been intertwined with gambling — ask the 1919 Chicago White Sox — and football in particular has attracted a fan base that increasingly seems to care less about who won or lost than about who covered the point spread. But only when the Big Dance rolls around every March do we see pretty much everyone who's watching the games, and a lot of people who aren't, attempt to predict an entire tournament. For instance, take this guy:

This fellow is "filling out a bracket," that is, forecasting the results not only of the 32 scheduled games but also of the 31 hypothetical games that would follow in the subsequent five rounds should all his picks turn out to be correct. As noted, this ritual has spread far beyond the ranks of college basketball fans; it's very common for a workplace to have an "office pool" in which everyone, including the stereotypical "secretary who makes picks based on which mascot sounds tougher," kicks in a couple of bucks and submits a bracket. There are a number of factors that make bracketology appealing even to civilians. One is that even people who don't much care about sports do tend to hold opinions about the universities in the tournament, or at the very least, about the parts of the country they represent. (Elizabeth's interest in basketball is roughly equivalent to my interest in shoes, yet we were able to spin a few minutes of conversation out of the all-important question of whether northern Iowa is better than Las Vegas.) Another is that completing a bracket is a sort of world-building exercise — a very limited one, yes, but still, by the time you get to the later rounds you've spun out this whole hypothetical future. Entering your bracket in an office pool is not entirely dissimilar to showing off your Quizilla results on Livejournal. Except that instead of having, like, six possible outcomes, you have over nine quintillion.

But I think the main thing is that, in filling out a bracket, you're not just wagering on a game — you're playing one. There's strategy involved. One of the fun things about the tournament is that the vast majority of the games are not even-strength: there's a clear favorite and a clear underdog, and each team receives a seeding that indicates where it falls in its region's pecking order. (This is yet another thing that makes it friendly to outsiders, since they can always just play the odds.) So you look at that matchup between (4) Vanderbilt and (13) Murray State and you think, hmm, yes, Vanderbilt's probably going to win, but Murray State probably has a better shot than the seedings would indicate... should I go for the more likely outcome, or pick the upset and hope to score some bonus points? And the answer to that question depends on the rule set of the pool you're in. Which brings me to my topic for today: what's the best way to score an NCAA tournament pool?

Obviously, it depends on what you want to reward. For instance, is it more important to pick a lot of early games correctly, or to nail the really important games at the end of the tournament? Many pools double the number of points awarded per game with each successive round: 1-2-4-8-16-32. This is how it works on, and there are a couple of rationales for it. One is that in any given first-round matchup there are only two possible winners, while in the championship game there are 64 possible candidates; since your task is 32 times harder, you should get 32 times as many points. Another is simply that the whole point of the tournament is to anoint a champion, so if you don't even pick the champion correctly there's no way you should win a pool. However, not everyone agrees with this. The first tournament pool I ever saw set the number of points awarded for each game equal to the round it was in: 1-2-3-4-5-6. To see the effect of this, compare the following two brackets. The first, marked "A.H.," is that of a Chicago-area teenager who claimed to have a perfect bracket after the first two rounds, but whose subsequent picks were terrible: he failed to correctly forecast a single Final Four team. The second, marked "P.B.," is also an actual bracket, this one belonging to someone whose first two rounds were fairly average (21 out of 32 correct picks in the first round, 9 out of 16 in the second) but who nailed three of the Final Four teams and the eventual winner. Here's how each would have fared under each system:

Round: 1st2nd3rd 4th5th6thTOTAL
A.H., 1-2-3-4-5-6 3232 15000 79
P.B., 1-2-3-4-5-6 2118 1212 56 74
A.H., 1-2-4-8-16-32 3232 20000 84
P.B., 1-2-4-8-16-32 2118 1624 1632 127

Since I think P.B. should win in this scenario, I prefer the 1-2-4-8-16-32 system. But I don't actually like either of the above rule sets, because neither one accounts for differences in team strength. Which brings me to the second consideration: how much to reward those who correctly pick surprising outcomes. According to the odds I was able to find online, if you were to have bet $100 on overwhelming underdog (14) Ohio to upset (3) Georgetown, you would have won $1100. Bet the same amount on (1) Duke, the eventual champion, to defeat (16) Arkansas-Pine Bluff, which started the season 0-11, and you would have won a grand total of $1.43. And yet standard practice in pools is to award these picks the exact same number of points, or at best, to award double points to an upset pick. Preposterous! Upsets are the soul of the tournament. Anyone can pick chalk; the laurels should go to those who identify which teams will outperform their seeds. And that's why every year I enter a contest that multiplies the points awarded per round by the seed of the winning team. Pick (1) Kansas to beat (16) Lehigh in the first round and you get a point; pick (11) Old Dominion to topple (6) Notre Dame and you get 11. Of course, this rule runs the risk of rewarding an anti-chalk strategy: why not predict that every first-round game will be an upset, and clean up? This is where the 1-2-4-8-16-32 multiplier becomes especially important. Even in an upset-heavy year like 2010, choosing all underdogs in the first round would have left you with no points in rounds three through six. And while those 12 points for picking (12) UTEP over (5) Butler look pretty sweet, they pale in comparison to the 155 points you get for picking Butler to win five games. Especially since those 155 points actually would have been awarded and the 12 would not have been.

Let's run some more numbers. In the table below, "chalk" indicates a strategy of always picking favorites; "anti-chalk" indicates a strategy of picking all underdogs in the first round, then playing favorites from there (since 9-12 seeds make it to the third round way more often than 13-16 seeds do). The system has no weighting by seed, while the system uses the scheme described above. Finally, "G.M." is a real bracket from 2009 that got two of the eventual Final Four teams, (1) North Carolina and (3) Villanova. A.H. and P.B. you already know, though I should probably specify that the three Final Four teams P.B. correctly picked were (1) Duke, (2) West Virginia, and, yes, (5) Butler. Here we go:

Round: 1st2nd3rd 4th5th6thTOTAL chalk 2009 2328 2416 1632 139 anti-chalk 2009 92 0000 11 G.M. 2009 2124 2016 1632 129 chalk 2009 8264 3216 1632 242 anti-chalk 2009 10824 0000 132 G.M. 2009 8954 2832 1632 251 chalk 2010 2216 168 00 62 anti-chalk 2010 106 0000 16 A.H. 2010 3232 20000 84 P.B. 2010 2118 1624 1632 127 chalk 2010 8332 248 00 147 anti-chalk 2010 10984 0000 193 A.H. 2010 192160 56000 408 P.B. 2010 8354 3664 1632 285

As you can see, the system favors chalk over anti-chalk even in a topsy-turvy year. Troublingly, in a more straightforward year, it favors chalk over someone who pegs a relatively big surprise (Villanova winning four rounds instead of two). While the system gives A.H. too much credit for picking early upsets and not enough of a penalty for missing the Butler story, it does a nice job of rewarding a balanced bracket over both chalk and anti-chalk.

In fact, I'd always considered the system pretty much perfect, but this year's tournament raised an interesting objection. Consider the NFL playoffs, which use what's called a "floating bracket." In each conference, six teams advance to the playoffs. In the first round, #3 plays #6, and #4 plays #5. The next weekend, #1 plays the weakest team that remains, whether that's #4, #5, or #6. But NCAA brackets are fixed. If a 1-seed makes it past the 16-seed, as has happened in every 1-16 pairing ever played, it goes on to play the 8-9 winner, even if weaker teams remain in the field. The flip side of this is that the winners of big first-round upsets play weaker teams in the second round than do the winners of small first-round upsets. So after (12) Cornell beat (5) Temple, it only had to defeat (4) Wisconsin to make it to the third round. But after (9) Northern Iowa defeated (8) UNLV, it had to play mighty (1) Kansas, a much more daunting task. When Cornell won its second-round game, it was surprising, but not overwhelmingly so: of the 4.8 million people who entered brackets in the contest, 11.3% had Cornell advancing to the Sweet Sixteen. But when Northern Iowa won its second-round game, dismissing the overall favorite, it was shocking. Only 0.9% of entrants had tabbed the Panthers to knock out the Jayhawks. So why did Cornell's victory merit 24 points and Northern Iowa's only 18?

This got me thinking: what if each game were worth a set number of points, using the 1-2-4-8-16-32 system, but then those points were divided up among all entrants who predicted a given winner? So right now I'm looking at a pool on — not the one I play in — with 29 entrants. All 29 picked Syracuse to win its second-round game; for this each would receive 2/29 or 0.069 points. Five people said Washington would do the same; they would receive 2/5 or 0.400 points. And the single brave soul who had St. Mary's winning a second-round game would get a full 2/1 or 2.000 points for that prediction. This seemed kind of cool, but I didn't like the way the decimals got really small. Then it occurred to me that you could achieve the same effect by taking the number of points corresponding to a given round and dividing by the percentage of people who had chosen a particular outcome. In this pool that would mean 2 for Syracuse, 11.6 for Washington, and a big 58 for St. Mary's.

Now, I imagine that such a system would encourage people to take more oddball choices in hopes of being one of the select few to benefit from them, which could really distort the percentages in small pools. In large ones, though, I think the effect would be positive. Michael Lewis has been promoting his new book The Big Short all over the place, and while I haven't read it, I've heard quite a few of his interviews. He says that one of the key concepts in the book is that people tend to drastically underestimate the likelihood of unlikely events. "Lehman Brothers collapse? That'll never happen!" Well, no — it might be unlikely to happen, but it could, and that means you probably shouldn't stake your entire financial system on the firm's survival. To return to basketball, take the 1993 NBA draft lottery. The system in place at the time took the eleven teams that didn't make the playoffs and assigned each one a number of ping-pong balls: eleven for the worst team, ten for the second-worst, and so on up to the best non-qualifying team, which got only one. In 1993, that team was the 41-41 Orlando Magic — and it won. Onlookers gasped, Bob Costas declared the result "unbelievable!", and Magic general manager Pat Williams called it "an absolute miracle." A one-in-66 chance is a longshot, certainly, but a miracle? A miracle is your pants spontaneously turning to gold, not "hey, your birthday is the same week as mine." Another example: the 2009-10 NBA season is currently drawing to a close, and I thought I'd compare the near-final standings to the predictions of ESPN's team of analysts. First up: the Milwaukee Bucks, currently ensconced in 6th place in the East after spending a fair amount of time in 5th. What did the ten prognosticators selected by the Worldwide Leader foresee? 13th, 14th, 14th, 14th, 14th, 14th, 15th, 15th, 15th, 15th. Not one was willing to take a flyer on the Bucks? Now let's look at the team sitting in 14th as I type this, the Washington Wizards. What said the prophets? 4th, 5th, 6th, 6th, 6th, 6th, 6th, 7th, 8th, 9th. A bit more of a range, but still closely clustered around 6th, and all completely wrong. How about out west, where the Oklahoma City Zombie Sonics hold down the 6th spot? ESPN's finest basketball minds came up with 9th, 10th, 11th, 11th, 11th, 11th, 11th, 12th, 12th, and 13th.

There was a funny bit in a Dollhouse episode in which Topher, needing to be in two places at once, uploads his mind into Victor's body and thereby creates a mental clone of himself. After their caper is over and they reconvene, Victor asks to keep Topher's brain patterns so he can help with the project and offer a second expert opinion. Topher replies: "It wouldn't be a second opinion! It'd be the same opinion twice!" What's the point of bringing together a huge panel if it's just going to be an echo chamber? You can sit there like a Bush Administration official and say, "But no one could have predicted that Andrew Bogut would start playing like a #1 overall pick and that Brandon Jennings would be a contender for rookie of the year!" and "No one could have predicted that Gilbert Arenas would start waving guns around and get suspended for the year!" But there are always a couple of teams that do much better than expected and there are always a couple of teams that suffer an unexpected collapse. The whole point of a prediction panel should be to guess which teams those will be! And the same is true for the NCAA tournament. Was it more likely that Kansas would make the final than that Butler would? Sure. But it was not, as the entrants seem to have thought, 198 times more likely. So I'm all for a system that discourages groupthink.

There is of course no way of telling what the percentages would look like for a contest such as the one I'm talking about. But using the numbers, here are some of the point totals that would have been awarded in 2009:

Round: 1st2nd3rd 4th5th6th
(1) North Carolina 1.012.11 4.7713.2 39.3118
(2) Michigan State 1.012.48 8.7559.3 2710
(3) Villanova 1.022.70 10.666.1 00
(3) Missouri 1.032.96 22.0000
(12) Arizona 2.2522.5 0000
(13) Cleveland State 14.900 000
(4) Wake Forest 000 000

You can make a good argument that this system rewards someone who makes the incorrect prediction that the result of the championship game would be, say, Michigan State over Pittsburgh, more than it rewards someone who makes the somewhat more accurate prediction that it would be North Carolina over Connecticut. That's why I said that what you consider the best system depends on your priorities. North Carolina was supposed to win six games. It did win six. And yes, someone who predicted that deserves a fair number of points for not forecasting an upset that never materialized. But Michigan State was only supposed to win three games, and it won five. Predicting that is, to me, a more impressive feat.

Now compare these numbers to those available in the more exciting tournament of 2010:

Round: 1st2nd3rd 4th5th6th
(1) Duke 1.012.36 5.4318.9 94.7500
(5) Butler 1.425.05 70.2500 53300
(5) Michigan State 1.113.88 83.3381 00
(6) Tennessee 1.3210.1 57.1000
(9) Northern Iowa 1.92222 0000
(10) St. Mary's 2.3454.1 0000
(11) Washington 3.8622.7 0000
(12) Cornell 3.0317.7 0000
(11) Old Dominion 5.8100 000
(14) Ohio 41.700 000
(1) Kansas 1.0000 000
(3) Georgetown 000 000

This isn't perfect either, but I like it. It rewards people for predicting the key stories of the tournament. In 2010, one of those stories was that Kansas was supposed to win six games and won one. Putting the correct winners where most people penciled in Kansas earned huge bonus points — Northern Iowa's eye-popping 222 is the obvious one, but look at how Michigan State goes from 3.88 for advancing to the Sweet Sixteen to 83.3 for continuing into the Elite Eight. You'd get more than twice as many points for choosing Tennessee to do the former, but only two-thirds as many for choosing Tennessee to do the latter... because Kansas wasn't in Tennessee's octet! Another big story, of course, was Butler. Butler went into the tournament as the #8 team in the country in the ESPN/USA Today poll, but out of 4.8 million entries in the contest, only 12,316 put it in the championship game. Now, did someone who predicted a final of Butler over East Tennessee State deserve to beat someone who, like P.B., picked the winner and put Butler in the Final Four, but didn't put Butler in the championship game? Mm, maybe not. So that 5330 might be pushing it. But note that it too is a product of Kansas Effect. Putting Butler in the Final Four only gets you as many points as picking 1-seed Duke to win the whole thing. But putting Butler in the championship game, well, that would mean picking against Kansas! Well over half the entrants — 59.3%! — had Kansas there. Perhaps more would have gone for Butler had they known that Butler would earn them 5330 points and Kansas only 27. And, as discussed, I would consider that side effect a big plus.

To close, here's how this system would have calculated the scores of those brackets covered above:

Round: 1st2nd3rd 4th5th6thTOTAL
chalk 2009 26.336.7 36.534.8 39.3118 292
anti-chalk 2009 41.022.5 0000 63.5
G.M. 2009 27.931.1 30.279.3 39.3118 326
chalk 2010 27.920.3 21.918.9 00 89.0
anti-chalk 2010 71.3317 0000 388
A.H. 2010 99.2367 86.6000 553
P.B. 2010 26.938.7 85.6546 94.7500 1290

And there you have it. Chalk beats anti-chalk in a boring year, but someone who backs the right underdog can still beat chalk: G.M. missed the bonus points associated with Michigan State, but even the Villanova points were enough to boost him past the all-favorites approach. In a more interesting year, anti-chalk beats chalk, but more importantly, someone who backs the right underdog beats both: P.B. missed the bonus points associated with putting Butler in the championship game, but even the Final Four points were enough to boost him past the gimmicks. And the fact that he was able to find the correct underdog at the end of the tournament rather than just during the first weekend gave him the edge over A.H. as well. So if I had to vote for a system it might well be this one.

Of course, all of this goes out the window next year when the NCAA destroys the tournament by going to 96 teams.

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